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Metamath Proof Explorer

Theorem joincl

Description: Closure of join of elements in the domain. (Contributed by NM, 12-Sep-2018)

Ref Expression
Hypotheses joincl.b 
|- B = ( Base ` K )
joincl.j 
|- .\/ = ( join ` K )
joincl.k 
|- ( ph -> K e. V )
joincl.x 
|- ( ph -> X e. B )
joincl.y 
|- ( ph -> Y e. B )
joincl.e 
|- ( ph -> <. X , Y >. e. dom .\/ )
Assertion joincl 
|- ( ph -> ( X .\/ Y ) e. B )

Proof

Step Hyp Ref Expression
1 joincl.b 
 |-  B = ( Base ` K )
2 joincl.j 
 |-  .\/ = ( join ` K )
3 joincl.k 
 |-  ( ph -> K e. V )
4 joincl.x 
 |-  ( ph -> X e. B )
5 joincl.y 
 |-  ( ph -> Y e. B )
6 joincl.e 
 |-  ( ph -> <. X , Y >. e. dom .\/ )
7 eqid 
 |-  ( lub ` K ) = ( lub ` K )
8 7 2 3 4 5 joinval 
 |-  ( ph -> ( X .\/ Y ) = ( ( lub ` K ) ` { X , Y } ) )
9 7 2 3 4 5 joindef 
 |-  ( ph -> ( <. X , Y >. e. dom .\/ <-> { X , Y } e. dom ( lub ` K ) ) )
10 6 9 mpbid 
 |-  ( ph -> { X , Y } e. dom ( lub ` K ) )
11 1 7 3 10 lubcl 
 |-  ( ph -> ( ( lub ` K ) ` { X , Y } ) e. B )
12 8 11 eqeltrd 
 |-  ( ph -> ( X .\/ Y ) e. B )